Gearing information for a Miele Umbria 100 bicycle

My bike is a Miele Umbria 100, either the 2005 or 2006 model (probably
2005 since I bought it early in 2006 and it was discounted). This is
a 21-speed bike; 3 front gears, 7 rear gears. Because I am interested
in such things, I have tried to work out the gear ratios and other
interesting information for it.

Miele's website for the current 2007
model Umbria 100 says that it has a 28/38/48 crank set (the front
gears) and a '14-28T' cog set (the rear gears). The only 7-speed rear
gear set that goes from 14 to 28 on Sheldon Brown's gear calculator is 14-16-18-20-22-24-28.
Since I have no desire to count gear teeth myself, I am going to
assume that both are accurate for my bike. I did check my lowest gear, which
is predicted to be 1:1 by this, and it is 1:1 or very close (one
wheel revolution in reverse forces one pedal revolution).

So, handy chart time. All of these use the gear notation that the bike's
shifters use, as opposed to the various confusing notations that bike
people like, and something like '1-2' has the front gear written first,
so it is the lowest front gear with the second lowest rear gear.

(When mapping between this and gear teeth, remember that the low gear
on the front chainring has the lowest number of teeth, but the low
gear on the rear chainring has the highest number of teeth. So my
lowest gear is 28 teeth to 28 teeth, for a 1:1 front to rear ratio.)

The gear ratios (front to rear); the gear combinations that the manual
says are okay to use are bolded.

front 1

front 2

front 3

rear 1

1.00

1.36

1.71

rear 2

1.17

1.58

2.00

rear 3

1.27

1.73

2.18

rear 4

1.40

1.90

2.40

rear 5

1.56

2.11

2.67

rear 6

1.75

2.38

3.00

rear 7

2.00

2.71

3.43

(Certain combinations of front and rear gears are to be avoided because the
chain runs at too much of an angle. This is how you turn a 21 speed bike
into a 13 speed one.)

Surprises: 2-6 and 3-4 are almost identical, but 1-4 and 2-2 are not.

The Umbria 100 gearing counts up straightforwardly for the gears you are
supposed to use, although it's not linear outside them; 2-1 is between
1-3 and 1-4, 2-7 is between 3-5 and 3-6, and 3-3 is between 2-5 and 2-6.
(All of these gears are usable for at least short periods of time.)

My bike computer doesn't have a cadence counter, but one can use its
km/h display to reverse engineer a brute force one. This chart maps
various pedal cadences in specific gears to the displayed km/h (using
the fact that I know what wheel size the bike computer is set to):

rpm:

30

40

50

60

70

80

90

100

110

120

1-1

3.9

5.2

6.5

7.8

9.1

10.3

11.6

12.9

14.2

15.5

1-2

4.5

6.0

7.5

9.1

10.6

12.1

13.6

15.1

16.6

18.1

1-3

4.9

6.6

8.2

9.9

11.5

13.2

14.8

16.5

18.1

19.7

1-4

5.4

7.2

9.1

10.9

12.7

14.5

16.3

18.1

19.9

21.7

2-2

6.1

8.2

10.2

12.3

14.3

16.4

18.4

20.5

22.5

24.6

2-3

6.7

8.9

11.2

13.4

15.6

17.9

20.1

22.3

24.6

26.8

2-4

7.4

9.8

12.3

14.7

17.2

19.7

22.1

24.6

27.0

29.5

2-5

8.2

10.9

13.6

16.4

19.1

21.8

24.6

27.3

30.0

32.8

2-6

9.2

12.3

15.4

18.4

21.5

24.6

27.6

30.7

33.8

36.9

3-4

9.3

12.4

15.5

18.6

21.7

24.8

27.9

31.0

34.1

37.2

3-5

10.3

13.8

17.2

20.7

24.1

27.6

31.0

34.5

37.9

41.4

3-6

11.6

15.5

19.4

23.3

27.2

31.0

34.9

38.8

42.7

46.5

3-7

13.3

17.7

22.2

26.6

31.0

35.5

39.9

44.3

48.8

53.2

(I am only bothering to list the gears I am supposed to use.)

Now, for my own use, the useful bits of the same information, namely
the speed bands for each gear and my target speed in the gear:

(gear)

(80 - 120)

(90)

1-1:

10.3 to 15.5

11.6

1-2:

12.1 to 18.1

13.6

1-3:

13.2 to 19.7

14.8

1-4:

14.5 to 21.7

16.3

2-2:

16.4 to 24.6

18.4

2-3:

17.9 to 26.8

20.1

2-4:

19.7 to 29.5

22.1

2-5:

21.8 to 32.8

24.6

2-6:

24.6 to 36.9

27.6

3-4:

24.8 to 37.2

27.9

3-5:

27.6 to 41.4

31.0

3-6:

31.0 to 46.5

34.9

3-7:

35.5 to 53.2

39.9

(Disclaimer: I may change my mind about my target cadences someday.)

Formulas

For my own reference if nothing else, here's how to calculate these.

Let:

fgt

be the number of teeth on the current front gear

rgt

be the number of teeth on the current rear gear

cad

be your pedaling cadence in RPM

circ

be the circumference of your wheel in millimeters
(as programmed into your bike computer)

Giving:

gr

fgt / rgt

the gear ratio, how many times the
rear wheel goes around when you pedal
through a complete circle.

mmm

gr * cad * circ

(the speed in millimeters per minute)

kmh

(mmm * 60) / (1000 * 1000)

the speed in km/h

Or, in fully worked out form:

kmh = (fgt * cad * circ * 60) / (rgt * 1000 * 1000)

I find it more convenient to do the km/h calculation in three steps,
because it makes it clearer where all the numbers are coming from.